PERIODIC MOTION 311 Making these substitutions In the left-hand member of the oreceding expression for the period we obtain = 4 = 4 I ffl mgD^Q L [by (XII)] [when a, is small higher! L terms may be neglected J when a is small sin^ = ~ • Therefore (XIV) 242. Simple Pendulum. — A ball which is suspended by means of a string forms a simple pendulum when it is free to swing about a horizontal axis through the upper end of the string, provided the mass of the string is negligible compared with that of the / ball and the radius of the ball is negli- <^ gible compared with the length of the string. If m denotes the mass of the ball and I the length of the string then -pIG 138 >rmg * This is called an elliptic integral. t This expansion is carried out by the Binomial theorem. See Appendix A- J See Appendix Ai.